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# HW4_soln - ECE 455 Optical Electronics Homework 4 Solutions...

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ECE 455: Optical Electronics Homework 4 Solutions 1 Lineshape I (a) The homogenous linewidth, which in this example consist totally of natural broadening, has contributions from both the upper and lower levels: Δ ν natural = 1 2 π 1 τ 1 + 1 τ 2 = 1 2 π 1 10 - 8 s + 1 3 × 10 - 9 s = 68 . 97 MHz ν 0 = E h = 0 . 5 eV · 1 . 602 × 10 - 19 J/eV 6 . 626 × 10 - 34 J - s = 1 . 209 × 10 14 s - 1 Δ ν doppler = 8 kTln 2 Mc 2 ! 1 / 2 ν 0 For Δ ν natural = Δ ν doppler we have: T = Δ ν natural ν 0 2 Mc 2 8 kln 2 = 6 . 897 × 10 7 s - 1 1 . 209 × 10 14 s - 1 ! 2 20 · 3 . 321 × 10 - 27 kg (3 × 10 8 m/s ) 2 8 · 1 . 38 × 10 - 23 J/K · ln 2 = 12 . 69 K (b) In order for the natural broadening to be equal to the Doppler broadening at room temperature, the natural linwidth must increase. This requires a much shorter lifetime for level 2 since the linewidth of the transition scales inversely with lifetime. Δ ν doppler = 8 kTln 2 Mc 2 ! 1 / 2 ν 0 = 8 · 1 . 38 × 10 - 23 J/K · 300 K · ln 2 20 · 1 . 67 × 10 - 27 kg · (3 × 10 8 m/s ) 2 !

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HW4_soln - ECE 455 Optical Electronics Homework 4 Solutions...

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